We will prove Theorem $2.7$ by filling in the blanks for the given two-column proof. Let's first recall what this theorem tells us.

Theorem $2.7$
Angles complementary to the same angle or to congruent angles are congruent.

This theorem can be divided into two parts: the given information and what we want to prove.

Given : Prove : ∠ 1 and ∠ 3 are complementary . ∠ 2 and ∠ 3 are complementary . ∠ 1 ≅ ∠ 2

Now, let's take a look at the statements that need to be completed one at a time.

Blank a.

The first blank asks us to state why

∠ 1

and

∠ 3,

as well as

∠ 2

and

∠ 3,

are complementary. That is part of the given information.

Statement : Reason : ∠ 1 and ∠ 3 are complementary ∠ 2 and ∠ 3 are complementary a . \underline{Given}

Blank b.

Now, let's take a look at the second blank. This wants to know why

m ∠ 1 + m ∠ 3 = 90

and

m ∠ 2 + m ∠ 3 = 90 .

This is from the definition of complementary angles.

Statement : Reason : m ∠ 1 + m ∠ 3 = 90 m ∠ 2 + m ∠ 3 = 90 b . \underline{Definition of comp . angles}

Blank c.

The third line in the proof table is looking at a new equation

m ∠ 1 + m ∠ 3 = m ∠ 2 + m ∠ 3 .

This is obtained by substituting the second equation from b into the first equation. Both equations are equal to

9 0^{\circ},

and therefore are equal to one another.

Statement : Reason : m ∠ 1 + m ∠ 3 = m ∠ 2 + m ∠ 3 c . \underline{Substitution Property}

Blank d.

The fourth blank wants to know what algebraic statement can be obtained by using the Reflexive Property. From part c, we can see that the common term on both sides is

\underline{m ∠ 3} .

Statement : Reason : d . \underline{m ∠ 3 = m ∠ 3} Reflexive Property

Blank e.

The next space asks why

m ∠ 1 = m ∠ 2 .

This is found by subtracting

m ∠ 3

from both sides of the equation in part c.

Statement : Reason : m ∠ 1 = m ∠ 2 e . \underline{Subtraction Property}

Blank f.

The sixth and final blank asks us why

∠ 1 ≅ ∠ 2 .

This is ultimately what we wanted to prove. This is obtained from the previous part by using the definition of congruent angles.

Statement : Reason : ∠ 1 ≅ ∠ 2 f . \underline{Definition of congruent angles}

Completed Proof

Below, we present the given table with all the blanks filled in.

Statements	Reasons
$∠ 1$ and $∠ 3$ are complementary $∠ 2$ and $∠ 3$ are complementary	a. Given
$m ∠ 1 + m ∠ 3 = 90$ $m ∠ 2 + m ∠ 3 = 90$	b. Definition of complementary angles
$m ∠ 1 + m ∠ 3$ $= m ∠ 2 + m ∠ 3$	c. Substitution Property of Equality
$\underline{m ∠ 3 = m ∠ 3}$	d. Reflexive Property of Equality
$m ∠ 1 = m ∠ 2$	e. Subtraction Property of Equality
$∠ 1 ≅ ∠ 2$	f. Definition of congruent angles

Exercise